If f(a)=K, then which of the following must be true?
f(K)=a
f inverse(k)=a
f inverse(a)=K
Properties of inverses:
If f(a)=K
then
1. f-1(K)=a
2. f-1(f(a))=a
... etc
None of the above answers is correct unless the second one says
f inverse(K)=a (i.e. uppercase K)
If ƒ(x ) = 2x, then ƒ -1(x ) =
If f(a) = K, then the statement "f inverse(K) = a" must be true. The inverse of a function swaps the roles of the input and output, so f inverse(K) would represent the input value that produces K as the output. Therefore, f inverse(K) = a.
To determine which of the given statements must be true, let's understand the concept of inverse functions and how they relate to the original function.
An inverse function, denoted as f^(-1), undoes the effects of the original function, f. In other words, if we have a function f that maps an input a to an output K, then the inverse function f^(-1) will map the output K back to the input a.
Now, let's evaluate each statement:
1. f(K) = a:
This statement suggests that if f(a) = K, then f(K) should equal a. However, this may not necessarily be true. The original function f(a) = K does not guarantee that f(K) = a. In fact, it's possible that f(K) is not defined or maps to a different value altogether. Therefore, statement 1 is not a must-be-true statement.
2. f^(-1)(K) = a:
This statement aligns with the definition of the inverse function. If f(a) = K, then f^(-1)(K) = a. The inverse function takes the output K and maps it back to the input a. Therefore, statement 2 must be true.
3. f^(-1)(a) = K:
This statement suggests that if f(a) = K, then f^(-1)(a) should equal K. However, this is not necessarily true. The inverse function does not necessarily map the input a to the original output K. It only maps the output K back to the input a. Therefore, statement 3 is not a must-be-true statement.
In conclusion, the statement that must be true is:
f^(-1)(K) = a